The Mixed Labeling Problem in Multi Target Particle Filtering

Transcription

1 The Mixed Labeling Problem in Multi Target Particle Filtering Yvo Boers Thales Nederland B.V. Haasbergerstraat 49, 7554 PA Hengelo, The Netherlands Hans Driessen Thales Nederland B.V. Haasbergerstraat 49, 7554 PA Hengelo, The Netherlands Abstract - In this paper the so called mixed labeling problem inherent, or at least thought to be inherent to a joint state multi target particle filter implementation is treated. The mixed labeling problem would be prohibitive for trac extraction from a joint state multi target particle filter. It is shown and proven using the theory of Marov chains, that the mixed labeling problem is inherently self-resolving in a particle filter. It is also shown that the factors influencing this capability are the number of particles and the number of resampling steps. Keywords: Particle Filters, Multi Target Tracing. Marov Processes. Introduction Recently quite an abundance of literature on multi target Bayesian filters has appeared. Using rigorous models, accounting for target birth, target death and closely spaced parallel moving objects necessitates the use of a staced or joint multi target state. See e.g. [] for a limited overview, quite some wors appeared even later, see e.g. [], [3], [4], these approaches all fall in the joint multi state class, where a particle represents a multi target state. We emphasize that approaches lie the e.g. the MC- JPDAF, see do not fall in this class, as they can be seen as a JPDAF filter, where the (E)KF part is replaced by a particle filter and do not employ a staced or joint multi state approach. An interesting problem is the so called mixed labeling problem inherent in a joint multi target density (JMTD) description approach, see e.g. []. This problem is thought to be prohibitive for the use of a particle filter based JMTD approach for trac extraction. In [] the problem is mentioned. But in a lot of wors the problem is not mentioned or maybe better ignored. This is partially because the problem will become prominent only in situations where objects move close and parallel for a substantial time period. In other words, for the problem to be prominent the a posteriori distributions on the different states must overlap significantly in all state parameters. Furthermore, if one is not interested in tracs per se, but e.g. only in the a posteriori density, then there is even no need to deal with problem. In this paper we will deal with this labeling problem and show its behaviour, especially in a particle filter setting. The contributions of this paper are:. Showing that the labeling problem is inherently self-resolving in a practical particle filter.. Rigorously prove this statement. 3. Show the ingredients that determine the speed of the self-resolution. 4. Illustrate the mixed labeling problem and its resolution in a simulation example using a multi target particle filter. The ey of the proof will turn out to rely on a property of a finite Marov process with absorbing states. More bacground on this topic is found in [6]. In [] it is argued that the joint multi state approach is invariant under permutations, loosely speaing this means that no distinction can be made between several labeling possibilities. Or equivalently, assuming a two target case, the filter cannot discern between or is invariant w.r.t. a situation in which trac one is associated with target one or trac one is associated with target two, where trac one refers to the upper part of the multi target state and trac two to the lower. Thus, s s and s s T T T T represent similar, symmetric or equal situations. Thus particles that differ only in the labeling are indistinguishable for the filter, they represent the same situation.

2 The above is also graphically illustrated in the next two figures by means of a colourful example. The particular scenario is not some much of interest now, but will become hereafter. Key is that two labeling choices are equivalent and indistinguishable for the filter. labeling, this is seen by the initially red sub-cloud on the left and the blue sub-cloud on the right, at a certain point the sub-clouds get mixed. This is the mixed labeling situation. Even though the filter was initially certain about the identity of the targets, it is confused now. If now, we were to use a straightforward trac extractor, i.e. just taing the mean over the multi target state, we would obtain the tracs as shown, that is we would end up somewhere in the middle of where the two targets actually are, again see Figure 3. So, the mixed labeling problem, combined with averaging as a trac extraction technique can potentially lead to a situation in which both targets are no longer described well by the tracs. We note, that this is the case even though the particle filter itself is still doing a good job in terms of describing the a posteriori multi target density. The problem described here, although focused on in the context of particle filters, is the exact same one that is at the root of the trac coalescence problem apparent in e.g. a standard PDAF filter, see [7] and references therein. Figure Labeling type one. Figure Labeling type two. Mixed labeling and particles In a JMTD filter, implemented through a particle filter every particle represents a hypothesis on a multi target state. Such a JMTD filter can be a filter in which a fixed, even nown, number of targets exist, but also a more involved setting in which target birth/death is accounted for and in which the number of targets is unnown and modeled as such. But even for the simplest problem, e.g. a two target setting, in which the two targets move closely and in parallel for some time the problem will occur. We have depicted such a scenario in Figure 3. In the scenario of Figure 3 two targets start out well separated and move closer and then parallel for quite a time, to separate again after some time. Even if we start a JMTD particle filter, where we start with only one type of Figure 3 Particles, mixed labeling and trac extraction. 3 Self resolving property of the JMTD In this section we will show, proof and illustrate that the label ambiguity problem solves itself over time in a JMTD. We will also indicate which elements are of importance for this self-resolving capacity. For the clarity of presentation we restrict ourselves mainly to two closely spaced targets, but the results are readily extended to the more general case. If appropriate we will indicate how the extension would be obtained. Let us assume we are in a situation lie the one depicted in Figure 3. That is we have two targets and therefore two labeling possibilities, let us say a labeling of the A type and a labeling of the B type. Also recall equations 0 and 0, indicating this. Thus every particle can be said either to be of the A or the B type.

3 Note that if we were to consider M targets we would have M! labeling possibilities. Now, assuming our JMTD filter has N particles under the two target hypothesis. One can even assume that it is nown to the filter that two targets exist, this will be of no importance to our argument to get to our result. Now, we A identify with the situation N particles of the A-type a state and we can thin of this as a state of time varying system. Note that this definition or use of state is different from the standard one we used for the inematic state of an object. Now realize that by doing so we obtain a finite Marov process with N + different states, where one can goes from one state to the other according to a Marov transition. Also observe that out of these N + states there are two end states or, see [6], two absorbing states, namely the two for which either N A A = N or N = 0. All other states are so called transient states. Thus, for the labeling we have a N + state Marov process with two absorbing states and N transient states. The Marov transition matrix of the above process is of the following form. Π 0 = A Marov process with a matrix of this form will end up in one of the absorbing states with probability one as time tends to infinity, see also [6]. The main consequence of all this, is that a mixed labeling situation is finally always resolved, thus we will not get stuc in a situation where we will have both A and B type particles, we always end up in a situation of only A or B particles. Thus, returning to the illustrative example of Figure 3, the situation therein cannot be a final or end situation, because clearly we are not in one of the absorbing states of our previously defined Marov process yet and we now that we must end up there!. Example: As an illustration consider the toy-example situation of only two particles. Each particle can have either a labeling A or B. Thus in the above context we have three states, namely AA, AB and BB. Let us furthermore assume that the probabilities for label switching are all equal. We then obtain the following Marov matrix: Π = 0 0 / 4 / / Starting in state AB, with probability one-half in the next time step one of the absorbing states (either AA or BB) is reached, where thereafter the system will remain of course in this state. Also for this toy example it is fairly easy e.g., to calculate the probability that after M steps the system has not reached an absorbing state yet. The reader may verify that this is probability is ( ) M. This result can be calculated directly or by the use of a fundamental Marov property, see [6], that the M-step transition probability, i.e. to be in state two after M-steps given that the starting state was state two, is obtained by the - element of the matrix: Π, thus looing at e the - M element of the Mth power of the Marov matrix. Thus relating this result to the original claim, that a process lie this will end in one of the absorbing states with probability one is also obvious now, as the probability that we will be in an absorbing state at time step M, provided we started in the transient one is: ( ) M, which clearly tends to for M to infinity. At this point, it is also in place to state clearly that the self-resolving property of the joint multi target particle filter is due to the fact that the number of particles is finite. This also means that in a practical situation with a large number of particles self resolution may tae a long time and cannot be waited for. Thus smart alternative methods for trac extraction would stile be needed. 4 Contributing factors It is easy to see that at least two factors contribute to the speed with which the labeling will be resolved over time. These are: The number of particles The degree of re-sampling The distribution of the particle weights The more particles, the more Marov states and thus the longer it taes to reach one of the absorbing ones, indicating unique labeling over all particles. Noting that a Marov transition occurs every time resampling is performed, it is obvious that re-sampling speeds up the process of reaching one of the absorbing states. If the weights are distributed in a non-uniform manner, or in other words are very sewed, the process of ending up in an end state of the Marov process is faster. It remains interesting to mae more quantitative statements w.r.t. the above contributing factors, this remains open for further investigation. Furthermore, at this point it is good to mention that in [], i.e. one of the few wors in which the labeling problem for the JMTD approach is mentioned and taen into

4 consideration, a K-means clustering approach is used to find that labeling that minimizes a K-means variance criterion. We emphasize that this a potentially dangerous thing to do, because it distorts the JMTD solution and steers towards minimal multi-modality, which might be unwanted, especially in the more interesting cases, where even a single target probability density might be multimodal, e.g. due to false alarms. Moreover, the specific type of filter implementation used in [], requires that mixed labeling must not occur at any time step and therefore the clustering is done at every time step. 5 Simulation Example In this section we will illustrate the phenomenon of mixed labeling and its self-resolving properties by means of an illustrative example. The example is a regular, that is plot based tracing situation of closely spaced targets. We have simulated a scenario, lie the one we have used in the previous sections. We employ a JMTD lie particle filter, lie e.g. in [] or [3]. for these alternative hypotheses are provided later on as well. In the next ten figures the particle cloud is shown at different time steps in the scenario. 0 0 Figure 4 Multi Target Density at Time Step 5. We jointly estimate the number of targets, the target states. In the scenario the targets move parallel for a long time period and in this time period their relative distance is small enough to cause the marginal a posterior distributions to have a significant overlap. The sensor is a sensor that produces Cartesian measurements and also has a non-unity detection probability, Pd=0.9. Furthermore, it is assumed that the number of false alarms follow a Poisson process with parameter λ V = 7, thus on average we receive 7 false alarms per scan uniformly distributed over our surveillance region. 0 We would lie to stress that even in a situation of no false alarmsλ V = 0 and no missed detections, Pd=, the labeling problem is relevant. 0 Figure 5 Multi Target Density at Time Step 5. During the entire scenario there are always two targets present. Furthermore, the sensor accuracies and the scenario is such that during a longer period, i.e. for about 0 scans, of the scenario the targets are moving close to one another and parallel at equal speeds. As already stated, they are close enough such that the marginal a posteriori probability densities significantly overlap during this period. We show the results at different time steps for a representative or typical realization of the scenario. For the sae of clarity we have only plotted the particles under the two targets hypothesis and not those under the zero and one target hypothesis. However, the probabilities

5 0 0 0 Figure 6 Multi Target Density at Time Step 7. 0 Figure 9 Multi Target Density at Time Step Figure 7 Multi Target Density at Time Step Figure 0 Multi Target Density at Time Step Figure 8 Multi Target Density at Time Step Figure Multi Target Density at Time Step 48.

6 0 and the fact that only particles are used help us here. In the next figure the filter output on the number of targets present is given. Note that during the entire scenario, the true number of targets present is two. The filter does not now this and has the ability to account for target birthdeath as well. It is in particular interesting to see that during the phase where the targets are close, the filter naturally is in doubt whether one or two targets are present, as one of them might have died off. After the targets have separated the filter becomes decisive again. 0.9 Probabilities for the number of targets 0 Figure Multi Target Density at Time Step # targets = 0 # targets = # targets = 0 Figure 3 Multi Target Density at Time Step 60. This scenario is a representative one to outline the mixed labeling problem. It can be seen, that during the period in which the targets are close, i.e. loo at Figure 6 through Figure 9, the sub clouds are mixed. During this period also physically the targets might be indistinguishable. At time step 46, see Figure 0, the cloud still has some mixed labeling, however after time step 48, see Figure, the mixed labeling is completely resolved. Although trac swap might, as it is unavoidable, occur, the labeling ambiguity over the particles is always resolved. We note that in this particular realization trac swap did not occur, this can be verified by the color coding on the different tracs. However, the probability of trac swap in this particular scenario is about 50%. This is because the targets are in quite close proximity over a long enough period. In this example, with particles in use, the mixed labeling is resolved remarably fast, without the use of extra trics or additional re-sampling. We do have to say that the fact that we perform resampling at every time step Figure 4 Mode probabilities on the number of targets present. (Truth= targets present during entire scenario.) 6 Conclusions It has been shown and proven that the Bayesian (particle based) JMTD approach to multi target tracing has the self-resolving capability with respect to the labeling problem. There are elements that can speed up the self resolving process, e.g. re-sampling. Once more we do stress however, that although a selfresolving property has been shown and proven, that this does not necessarily mean that in a practical setting, in which e.g. trac extraction is mandatory at all times, the mixed labeling problem is not causing problems. In fact it is and this problem is dealt with in more detail in an accompanying paper, see []. This is also a topic for ongoing and future research. Furthermore, as a topic for future wor it would be interesting to see whether more quantitative statistical statements could be made w.r.t. the self-resolving property. Acnowledgement This wor has been partly financially supported by the Swedish and Dutch MOD representatives, as a European MOU, ERG no. 0, research and technology project. This support is gratefully acnowledged.

7 7 References [] B. Ristic, S. Arulampalam and N. Gordon. Beyond the Kalman filter particle filters for tracing applications. Artech House, Boston, 004. [] C. Kreucher, K. Kastella and A.O. Hero. Multi Target Tracing using the Joint Multi Target Probability Density. IEEE Transactions on Aerospace and Electronic Systems [3] T. Vercauteren, D. Guo and X. Wang. Joint Multi Target Tracing and Classification in Collaborative Sensor Networs. IEEE Journal on selected areas in communications, vol. 3, no. 4, 005. [4] M.R. Morelande, C. Kreucher and K. Kastella. A Bayesian approach to multiple target tracing. Submitted to IEEE Transactions on Signal Processing, 006. [5] J. Vermaa, S.J. Godsill and Patric Perez. Monte Carlo Filtering for Multi Target Tracing and Data Association. IEEE Transactions on Aerospace and Electronic Systems [6] H.M. Taylor and S. Karlin. An Introduction to Stochastic Modeling. 3rd Edition. Academic Press, London, UK, 998. [7] H. A. P. Blom and E. A. Bloem. Joint particle filtering of multiple maneuvering targets from unassociated measurements. Journal of Advances in Information Fusion. Vol., No., 006. [8] Y. Boers and J.N. Driessen. A Particle Filter Multi Target Trac Before Detect Application. IEE Proceedings - Radar, Sonar and Navigation, vol. 5, no. 6, 004 [9] J.N. Driessen and Y. Boers. Numerically Effective Multiple Model Particle Filter Algorithms. IEE Proceedings - Radar, Sonar and Navigation, vol. 5, no. 5, 005. [0] S. Blacman and R. Popoli. Design and Analysis of Modern Tracing Systems. Artech, Norwood, MA, 999. [] M. Eman, Y. Boers, J.N. Driessen, E. Svistiens and L. Sjoberg. Particle Filters for Tracing Closely Spaced Targets. Submitted to Fusion 007.